Fault location during pole-open condition

ABSTRACT

The present disclosure illustrates the errors that are encountered when using both single-ended and double-ended normal-mode fault location calculations when a fault occurs in a pole-open condition. The disclosure provides systems and methods for accurately calculating the location of faults that occur during pole-open conditions, including single-ended approaches and double-ended approaches.

TECHNICAL FIELD

This disclosure relates to identifying fault locations on a transmissionline. More particularly, this disclosure provides systems and methodsfor determining a fault location during a pole-open condition.

BRIEF DESCRIPTION OF THE DRAWINGS

Non-limiting and non-exhaustive embodiments of the disclosure aredescribed herein, including various embodiments of the disclosure withreference to the figures listed below.

FIG. 1 illustrates a simplified diagram of a generic single-line powersystem with a fault at a location, d.

FIG. 2 illustrates a simplified diagram of a sequence network for aphase B pole-open condition, according to one embodiment.

FIG. 3 illustrates a simplified diagram of a sequence network for aphase B pole-open condition showing a phase-A-to-ground fault network,according to one embodiment.

FIG. 4 illustrates a simplified diagram of an example single-line powersystem with a fault at a location, d.

FIG. 5 illustrates a graph of the normal-mode current distributionfactors for the example single-line power system of FIG. 4.

FIG. 6 illustrates distance calculation errors when using normal-modedistance calculations with zero-sequence (solid line) andnegative-sequence (dashed line) polarization during the open-pole faultcondition in FIG. 4.

FIG. 7 illustrates that the reason that the normal-mode distancecalculation approaches in FIG. 6 for the single-line power system ofFIG. 4 are inaccurate is because the pre-fault zero-sequence andnegative-sequence currents are non-negligible.

FIG. 8 illustrates the variation of the current distribution factorphase angles for each of the zero-sequence, positive-sequence, andnegative-sequence currents for the open-pole fault condition in FIG. 4.

FIG. 9 illustrates a single-ended fault location calculation following aphase B pole-open condition using the two sequence network calculationapproach described herein.

FIG. 10 illustrates a graphical representation of a fault locationcalculation following a phase B pole-open condition using an iterativetwo sequence network calculation approach with a polynomialapproximation of the tilt angle.

FIG. 11 illustrates a simplified diagram of a negative-sequence network,according to one embodiment.

FIG. 12 illustrates a graphical representation of a fault locationcalculation of the single-line power system in FIG. 4 using adouble-ended normal-mode approach and the double-ended pole-openapproach (dual-sequence approach).

FIG. 13 illustrates a close-up view of FIG. 12 to better illustrate theaccuracy of the pole-open, dual-sequence approach.

FIG. 14 illustrates another example diagram of a single-line powersystem with a fault at a location d, according to one embodiment.

FIG. 15 illustrates another graphical representation showing theimproved accuracy of the pole-open, dual-sequence approach as comparedto the normal-mode approach.

FIG. 16 illustrates a close-up view of FIG. 15 to better illustrate theabsolute accuracy of the pole-open, dual-sequence approach.

FIG. 17 illustrates a flow chart of a method for dual-sequence,single-ended fault location calculation, according to one embodiment.

FIG. 18 illustrates a flow chart of a method for dual-sequence,double-ended fault location calculation, according to one embodiment.

FIG. 19 illustrates an IED for performing single-ended or double-endedfault location calculation, according to one embodiment.

In the following description, numerous specific details are provided fora thorough understanding of the various embodiments disclosed herein.The systems and methods disclosed herein can be practiced without one ormore of the specific details, or with other methods, components,materials, etc. In addition, in some cases, well-known structures,materials, or operations may not be shown or described in detail inorder to avoid obscuring aspects of the disclosure. Furthermore, thedescribed features, structures, or characteristics may be combined inany suitable manner in one or more alternative embodiments.

DETAILED DESCRIPTION

This disclosure provides methods and systems to accurately locate afault during a pole-open condition. Power distribution and protectionequipment may include fault location technology to identify the locationof a fault along a transmission line. Both single-ended and double-endedfault location determination equipment traditionally makes theassumption that all the three poles (i.e., phase lines) of a three-phasepower transmission system are closed. Traditional fault locationdetermination systems and methods use measured and calculated data froma single sequence network.

Conventional fault location calculations, whether based on single-endedor double-ended data, exhibit significant errors following a pole-opencondition because they rely on data from a single sequence network.These systems are particularly inaccurate if prefault sequence currents(negative or zero) are present, such as when the fault resistanceincreases so that the sequence current during the fault becomescomparable to the sequence current existing before the fault.

The present disclosure provides systems and methods that utilize thecombined information from two sequence networks, positive and negative,to achieve accuracy comparable to the normal-mode (three pole closed)conditions for both normal-mode and pole-open modes. Whereas the singlesequence network approach ignores the voltage difference across the openbreaker pole in a double-ended calculation, a dual-sequence networkapproach compensates for the voltage potential across the open breakerpole.

The fault location systems and methods describe herein utilize or mayutilize impedance-based techniques and process all six voltage andcurrent waveforms from a single end or both ends of a transmission lineand do not rely on wave-based fault location techniques.

In a single-ended fault location system, during a pole-open condition,the distance to the fault can be determined by using the voltages andcurrents on each phase. A voltage sensor may detect the voltage for thefaulted loop, VAL. A current sensor (e.g., a current transformer) maydetect a prefault zero-sequence current, I0L_(pref). A prefaultzero-sequence current may be determined during the pole-open conditionbefore any other fault occurs. I0L_(pref) may be stored in memory by afault location system. The current sensor may also detect azero-sequence current, I0L, during the fault and calculate a current forthe faulted loop, I_(AG).

Other values such as impedance and phase angles may be calculated,measured, and/or looked up in a table. For example, the linepositive-sequence impedance, ZL1, may be measured or calculated usingvoltage and/or current measurements. Finally, a current distributionfactor phase angle, e^(−jψ) ⁰ , may be calculated by using theimpedances of both the positive-sequence and zero-sequence networks.

Using these values, a single-ended fault location system may determinethe distance to a fault. The distance may be calculated by finding thedifference, ΔI0L, of I0L and I0L_(pref), multiplying the difference,ΔI0L, by e^(−jψ) ⁰ , and multiplying the conjugate of that product byVAL. The imaginary portion of the resulting product may be divided bythe imaginary portion of the product of ZL1, I_(AG), and the conjugateof the product of ΔI0L and e^(−jψ) ⁰ , such that:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {{VAL} \cdot {{conj}\left( {\Delta \; I\; 0{L \cdot e^{{- j}\; \psi_{0}}}} \right)}} \right\rbrack}{{Im}\left\lbrack {{ZL}\; {1 \cdot I_{AG} \cdot {{conj}\left( {\Delta \; I\; 0{L \cdot e^{{- j}\; \psi_{0}}}} \right)}}} \right\rbrack}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

Equation 1 is different from those currently used to find faults innormal mode with all three phases closed. One difference is that for apole-open condition, the incremental quantity of the zero-sequencecurrent may be used (i.e., the difference between the prefaultzero-sequence current existing during the pole-open condition and thezero-sequence current during/after the fault). The current distributionfactor phase angle may be found using the impedances of both thepositive-sequence and zero-sequence networks.

In a double-ended fault location system, during a pole-open conditionthe distance to the fault can be determined by using a compensated andcorrected equation. In a double-ended fault location system the voltagesand currents on each phase at both ends of a line may be used. Toaccurately locate the fault, the negative-sequence network may beexpressed as an equation. Additionally or alternatively, thepositive-sequence network may provide an equivalent sequence voltage.The equivalent sequence voltage may be extracted from thepositive-sequence network and be used to replace the sequence voltageterm in the negative-sequence network equation. For example, if phase Bis pole-open the distance may be expressed as:

$\begin{matrix}{d = \frac{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {{a \cdot I}\; 1R}} \right)} - {a\left( {{V\; 1L} - {V\; 1R}} \right)}}{{{ZL}\; 1\left( {{I\; 2L} + {I\; 2R}} \right)} - {{a \cdot {ZL}}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

As another example, if phase A was open the distance may be expressedas:

$\begin{matrix}{d = \frac{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {I\; 1R}} \right)} - \left( {{V\; 1L} - {V\; 1R}} \right)}{{{ZL}\; 1\left( {{I\; 2L} + {I\; 2R}} \right)} - {{ZL}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

As another example, if phase C was open the distance may be expressedas:

$\begin{matrix}{d = \frac{\begin{matrix}{\left( {{V\; 2L} - {V\; 2R}} \right) +} \\{{{ZL}\; 1\left( {{I\; 2R} - {{a^{2} \cdot I}\; 1R}} \right)} - {a^{2}\left( {{V\; 1L} - {V\; 1R}} \right)}}\end{matrix}}{{{ZL}\; 1\left( {{I\; 2L} + {I\; 2R}} \right)} - {{a^{2} \cdot {ZL}}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}} & {{Equation}\mspace{14mu} 4}\end{matrix}$

Additional details and examples are provided with reference to thefigures below. Generally speaking, the systems and methods disclosedherein may be adapted to interface with or be included as part of aprotection element or protection ecosystem, such as a power systemprotection relay. Such protection devices may be configured tocommunicate with, control, operate, energize, de-energize, and/ordisengage one or more power system components and provide an indicationof where a potential fault has occurred. Protection relays may beinstalled in electric power transmission and distribution facilities todetect overloads, short circuits, and other fault conditions.

Many embodiments of a protective relay include electronic devicesemploying FPGAs, microcontrollers, CPUs, A/D converters, electronicdisplays, communication ports, and other electronic devices and systemsto digitize power system current and/or voltage measurements, andprocess data associated with the digitized currents and voltagesaccording to various algorithms and/or digital filters. A protectiverelay may be configured to make protection decisions based on dataassociated with a digitized power system's currents and/or voltages, andmay communicate decisions made by a protective relay to an appropriatesystem or personnel, and/or may otherwise cause a suitable response tothe digitized power system's currents and/or voltages.

An intelligent electronic device (IED), which may be used formonitoring, protecting, and/or controlling industrial and utilityequipment, such as in electric power delivery systems may include systemcomponents to implement a method for identifying fault locations usingthe two sequence network approaches described herein. Such IEDs may beconfigured to use a single-ended two sequence network approach or adouble-ended two sequence network approach. In both of theseembodiments, the IED is able to provide accurate fault locationinformation even during a pole-open event.

The phrases “connected to” and “in communication with” refer to any formof interaction between two or more components, including mechanical,electrical, magnetic, and electromagnetic interaction. Two componentsmay be connected to each other, even though they are not in directcontact with each other, and even though there may be intermediarydevices between the two components.

As used herein, the term IED may refer to any microprocessor-baseddevice that monitors, controls, automates, and/or protects monitoredequipment within a system. Such devices may include, for example, remoteterminal units, differential relays, distance relays, directionalrelays, feeder relays, overcurrent relays, voltage regulator controls,voltage relays, breaker failure relays, generator relays, motor relays,automation controllers, bay controllers, meters, recloser controls,communications processors, computing platforms, programmable logiccontrollers (PLCs), programmable automation controllers, input andoutput modules, motor drives, and the like. IEDs may be connected to anetwork, and communication on the network may be facilitated bynetworking devices including, but not limited to, multiplexers, routers,hubs, gateways, firewalls, and switches. Furthermore, networking andcommunication devices may be incorporated in an IED or be incommunication with an IED. The term IED may be used interchangeably todescribe an individual IED or a system comprising multiple IEDs.

Some of the infrastructure that can be used with embodiments disclosedherein is already available, such as: general-purpose computers,computer programming tools and techniques, digital storage media, andcommunications networks. A computer may include a processor, such as amicroprocessor, microcontroller, logic circuitry, or the like. Theprocessor may include a special-purpose processing device, such as anASIC, PAL, PLA, PLD, CPLD, Field Programmable Gate Array (FPGA), orother customized or programmable device. The computer may also include acomputer-readable storage device, such as non-volatile memory, staticRAM, dynamic RAM, ROM, CD-ROM, disk, tape, magnetic, optical, flashmemory, or other computer-readable storage medium.

Suitable networks for configuration and/or use, as described herein,include any of a wide variety of network infrastructures. Specifically,a network may incorporate landlines, wireless communication, opticalconnections, various modulators, demodulators, small form-factorpluggable (SFP) transceivers, routers, hubs, switches, and/or othernetworking equipment.

The network may include communications or networking software, such assoftware available from Novell, Microsoft, Artisoft, and other vendors,and may operate using TCP/IP, SPX, IPX, SONET, and other protocols overtwisted pair, coaxial, or optical fiber cables, telephone lines,satellites, microwave relays, modulated AC power lines, physical mediatransfer, wireless radio links, and/or other data transmission “wires.”The network may encompass smaller networks and/or be connectable toother networks through a gateway or similar mechanism.

Aspects of certain embodiments described herein may be implemented assoftware modules or components. As used herein, a software module orcomponent may include any type of computer instruction orcomputer-executable code located within or on a computer-readablestorage medium, such as a non-transitory computer-readable medium. Asoftware module may, for instance, comprise one or more physical orlogical blocks of computer instructions, which may be organized as aroutine, program, object, component, data structure, etc., that performone or more tasks or implement particular data types, algorithms, and/ormethods.

A particular software module may comprise disparate instructions storedin different locations of a computer-readable storage medium, whichtogether implement the described functionality of the module. Indeed, amodule may comprise a single instruction or many instructions, and maybe distributed over several different code segments, among differentprograms, and across several computer-readable storage media. Someembodiments may be practiced in a distributed computing environmentwhere tasks are performed by a remote processing device linked through acommunications network. In a distributed computing environment, softwaremodules may be located in local and/or remote computer-readable storagemedia. In addition, data being tied or rendered together in a databaserecord may be resident in the same computer-readable storage medium, oracross several computer-readable storage media, and may be linkedtogether in fields of a record in a database across a network.

The embodiments of the disclosure can be understood by reference to thedrawings, wherein like parts are designated by like numerals throughout.The components of the disclosed embodiments, as generally described andillustrated in the figures herein, could be arranged and designed in awide variety of different configurations. Thus, the following detaileddescription of the embodiments of the systems and methods of thedisclosure is not intended to limit the scope of the disclosure, asclaimed, but is merely representative of possible embodiments.

FIG. 1 illustrates a simplified diagram of single-line power system 100with a fault at a location, d. The single-line power system is modeledas a voltage VS 110 connected to a transmission line with an impedanceZS1/ZS0 111 between the power source VS 110 and a left bus L 112. Thetransmission line may have an impedance ZL1/ZL0 113 between the left busL 112 and a right bus R 114. A fault location 113 a distance d from theleft bus L 112 is shown as well. The transmission line may have animpedance ZR1/ZR0 115 between the right bus R 114 and a voltage VT 120.

The apparent loop impedance, Z_(LP), as seen from the left bus L 112 forany of the three ground fault loops and three phase fault loops can beexpressed by the following equation with reference to FIG. 1:

$\begin{matrix}{Z_{LP} = {\frac{V_{LP}}{I_{LP}} = {{{d \cdot {ZL}}\; 1} + {{Rf}\; \frac{K_{LP}}{K_{I}}}}}} & {{Equation}\mspace{14mu} 5}\end{matrix}$

V_(LP) and I_(LP) represent the voltage and current relative to theparticular faulted loop with reference to FIG. 1. The distance, d, inEquation 5 and FIG. 1 is the distance to the fault in per unit (pu) linelength. ZL1 is the line positive-sequence impedance and Rf is the faultresistance. K_(LP) is a network parameter and K is the ratio of thefaulted loop current over the change of the same current. Voltage andcurrent values corresponding to various K_(LP) factors are described in“Tutorial on the Impact of Network Parameters on Distance ElementResistance Coverage” by G. Benmouyal, A Guzman, and R. Jain, publishedin the 40th Annual Western Protective Relay Conference, Spokane, Wash.,October 2013.

With reference to FIG. 1 and Equation 5 above, the following variablesare defined:

$\begin{matrix}{{V_{LP} = {V_{AG} = {VAL}}}{I_{LP} = {I_{AG} = {{IAL} + {{K_{0} \cdot I}\; 0L}}}}} & {{Equation}\mspace{14mu} 6} \\{K_{LP} = {K_{AG} = \frac{3}{{2\; {Cl}} + {{C0}\left( {1 + K_{0}} \right)}}}} & {{Equation}\mspace{14mu} 7} \\{K_{I} = {\frac{I_{LP}}{I_{LP} - I_{LD}} = {\frac{I_{LP}}{\Delta \; I_{LP}} = \frac{\left( {{IAL} + {{K_{0} \cdot I}\; 0L}} \right)}{\left( {{IAL} + {{K_{0} \cdot I}\; 0L}} \right) - I_{LD}}}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

The zero-sequence compensation factor for Equation 6 above is definedas:

$\begin{matrix}{K_{0} = \frac{{{ZL}\; 0} - {{ZL}\; 1}}{{ZL}\; 1}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

In Equation 7 above, C1 is the positive-sequence current distributionfactor as seen from the left bus L 112. C2 is the negative-sequencecurrent distribution factor and is equal to C1. C0 is the zero-sequencenetwork current distribution factor. All three quantities are providedas:

$\begin{matrix}{{C\; 1} = {{C\; 2} = \frac{{\left( {1 - d} \right){ZL}\; 1} + {{ZR}\; 1}}{{{ZL}\; 1} + {{ZS}\; 1} + {{ZR}\; 1}}}} & {{Equation}\mspace{14mu} 10} \\{{C\; 0} = \frac{{\left( {1 - d} \right){ZL}\; 0} + {{ZR}\; 0}}{{{ZL}\; 0} + {{ZS}\; 0} + {{ZR}\; 0}}} & {{Equation}\mspace{14mu} 11}\end{matrix}$

In Equation 8 above, I_(LD) is the load or the prefault loop current.For a phase-A-to-ground fault, the prefault current is calculated as:

$\begin{matrix}{I_{{AG}\_ {pf}} = \frac{{VS} - {VT}}{{{ZL}\; 1} + {{ZS}\; 1} + {{ZR}\; 1}}} & {{Equation}\mspace{14mu} 12}\end{matrix}$

According to the Takagi principle, the normal-mode, single-sequencedistance calculation is based on the voltage of the primary loop asfollows:

V _(LP) =d·ZL1·I _(LP) +Rf·K _(LP)(I _(LP) −I _(LD))  Equation 13

The normal-mode, single-ended fault location calculation using theTakagi principle makes the assumption that the K_(LP) is a pure realnumber which results in a distance equation:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {V_{LP} \cdot {{conj}\left( {\Delta \; I_{LP}} \right)}} \right\rbrack}{{Im}\left\lbrack {{ZL}\; {1 \cdot I_{LP} \cdot {{conj}\left( {\Delta \; I_{LP}} \right)}}} \right\rbrack}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

A Takagi method for finding the fault current distance, d, issusceptible to errors because of the assumption that the factor K_(LP)is a real number. One approach to compensate for these errors includesintroducing a tilt angle, θ, (e.g., a tilt factor or tilt value) tocompensate for the phase angle factor that is neglected in Equation 14.The distance, d, to the fault according to such an embodiment isexpressed below:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {V_{LP} \cdot {{conj}\left( {\Delta \; {I_{LP} \cdot e^{j\; \theta}}} \right)}} \right\rbrack}{{Im}\left\lbrack {{ZL}\; {1 \cdot I_{LP} \cdot {{conj}\left( {\Delta \; {I_{LP} \cdot e^{j\; \theta}}} \right)}}} \right\rbrack}} & {{Equation}\mspace{14mu} 15}\end{matrix}$

The value of the tilt angle, θ, is a function of the distance, d, to thefault. If an accurate knowledge of the network impedances is available,an iterative approach can be used to solve for the title angle, θ. Inother embodiments, a single value for the tilt angle, θ, is used for theentire range of the distance, d.

For certain fault types, such as single-phase-to-ground faults, it ispossible to eliminate the prefault current by expressing the voltage ofthe faulted impedance loop as a function of the total current at thefault, as expressed below:

V _(AG) =d·ZL ₁ ·I _(AG)+3Rf·I1F  Equation 16

For single phase-A-to-ground faults, the fault location for the totalsequence currents can be expressed as:

I1F=I2F=I0F  Equation 17

The negative- and zero-sequence currents from the left bus L 112 can beexpressed as functions of the total negative-sequence and zero-sequencecurrents at the fault as follows:

I2L=C2·I2F

I0L=C0·I0F  Equation 18

Accordingly, the voltage of the faulted impedance loop for thephase-A-to-ground fault can be expressed as a function of thezero-sequence current:

$\begin{matrix}{V_{AG} = {{{d \cdot {ZL}}\; {1 \cdot I_{AG}}} + {3\; {{Rf} \cdot \frac{I\; 0L}{C0}}}}} & {{Equation}\mspace{14mu} 19}\end{matrix}$

Using the identity:

${{Im}\left( \frac{I\; 0\; {L \cdot {{conj}\left( {I\; 0{L \cdot e^{{- j}\; \varphi_{0}}}} \right)}}}{{{C\; 0}} \cdot e^{j\; \varphi_{0}}} \right)} = {{{Im}\left( \frac{I\; 0L^{2}}{{C0}} \right)} = 0}$

The distance to the fault can be expressed as:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {V_{AG} \cdot {{conj}\left( {I\; 0\; {L \cdot e^{{- j}\; \varphi_{0}}}} \right)}} \right\rbrack}{{Im}\left\lbrack {{ZL}\; {1 \cdot I_{AG} \cdot {{conj}\left( {I\; 0\; {L \cdot e^{{- j}\; \varphi_{0}}}} \right)}}} \right\rbrack}} & {{Equation}\mspace{14mu} 20}\end{matrix}$

The same reasoning can be applied using the negative-sequence current todevelop the equation:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {V_{AG} \cdot {{conj}\left( {I\; 2{L \cdot e^{{- j}\; \varphi_{2}}}} \right)}} \right\rbrack}{{Im}\left\lbrack {{ZL}\; {1 \cdot I_{AG} \cdot {{conj}\left( {I\; 2\; {L \cdot e^{{- j}\; \varphi_{2}}}} \right)}}} \right\rbrack}} & {{Equation}\mspace{14mu} 21}\end{matrix}$

Equivalent expressions can be made for phase-B-to-ground orphase-C-to-ground faults by introducing the corresponding loop voltagesand currents.

The equations above, in the context of the problem shown in FIG. 1, areaccurate for normal-mode events where all three phases are closed.However, the equations and approaches described above are susceptible tosignificant errors in the calculation of the distance, d, to the faultduring a pole-open fault event.

Specifically, Equation 18 above accurately represents the sequencecurrents during a normal-mode fault event (i.e., an event that occurswhen all three phases are closed). However, during a pole-open faultevent, Equation 18 does not accurately represent the sequence currents.

To demonstrate this shortcoming of the normal-mode approach to apole-open fault event, the analysis below uses the single-line diagramof FIG. 1 and assumed a phase-B pole-open condition.

FIG. 2 illustrates a simplified diagram of a sequence network 200 for aphase B pole-open condition, according to one embodiment. Expressing thepole-open condition as phase B being open between two points, x and y,the sequence voltages can be expressed as follows:

$\begin{matrix}{\begin{pmatrix}{V\; 1_{xy}} \\{V\; 2_{xy}} \\{V\; 0_{xy}}\end{pmatrix} = {\frac{1}{3} \cdot \begin{pmatrix}1 & a & a^{2} \\1 & a^{2} & a \\1 & 1 & 1\end{pmatrix} \cdot \begin{pmatrix}{VA}_{xy} \\{VB}_{xy} \\{VC}_{xy}\end{pmatrix}}} & {{Equation}\mspace{14mu} 22}\end{matrix}$

In Equation 22, a is the conventional complex operator 1∠120°. SinceVA_(xy) and VC_(xy) are zero, the sequences can be expressed as:

$\begin{matrix}{{{V\; 1_{xy}} = {\left( \frac{1}{3} \right){aVB}_{xy}}}{{V\; 2_{xy}} = {\left( \frac{1}{3} \right)a^{2}{VB}_{xy}}}{{V\; 0_{xy}} = {\left( \frac{1}{3} \right){VB}_{xy}}}} & {{Equation}\mspace{14mu} 23}\end{matrix}$

The current constraint is expressed by the condition that the phase Bcurrent must be equal to zero or:

IB=a ² I1L+aI2L+I0L=0  Equation 24

A plurality ideal transformers 212, 222, and 232 represented in thesequence network of FIG. 2 implement the two voltage and currentconstraints for a positive-sequence loop 210, a negative-sequence loop220, and a zero-sequence loop 230.

The sequence network of the phase B pole-open condition of FIG. 2 isprovided below:

$\begin{matrix}{{\begin{pmatrix}\begin{matrix}{{{ZL}\; 1} +} \\{{{ZS}\; 1} + {{ZR}\; 1}}\end{matrix} & 0 & 0 & a \\0 & \begin{matrix}{{{ZL}\; 1} +} \\{{{ZS}\; 1} + {{ZR}\; 1}}\end{matrix} & 0 & a^{2} \\0 & 0 & \begin{matrix}{{{ZL}\; 0} +} \\{{{ZS}\; 0} + {{ZR}\; 0}}\end{matrix} & 1 \\a^{2} & a & 1 & 0\end{pmatrix} \cdot \begin{pmatrix}{I\; 1L} \\{I\; 2L} \\{I\; 0L} \\{\left( \frac{1}{3} \right){VB}_{xy}}\end{pmatrix}} = \begin{pmatrix}{{VL} - {VR}} \\0 \\0 \\0\end{pmatrix}} & {{Equation}\mspace{14mu} 25}\end{matrix}$

Using the following defined variables, the sequence currents of thesequence network of FIG. 2 can be determined:

ΔV=VS−VT

m=ZL1+ZS1+ZR1=ZL2+ZS2+ZR2

n=ZL0+ZS0+ZR0

m ₁=−(1−d)ZL1−ZR1

n ₁=−(1−d)ZL0−ZR0

p=m+m ₁ =dZL1+ZS1=dZL2+ZS2

q=n+n ₁ =dZL0+ZS0=dZL0+ZS0  Equation 26

Using a Gaussian elimination process, the three prefault sequencecurrents can be calculated as follows:

$\begin{matrix}{{I\; 1L_{{pref}\; 1\; t}} = {{{- I}\; 1R_{{pref}\; 1\; t}} = \frac{\Delta \; {V\left( {m + n} \right)}}{m\left( {m + {2n}} \right)}}} & {{Equation}\mspace{14mu} 27} \\{{I\; 2L_{{pref}\; 1\; t}} = {{{- I}\; 2R_{{pref}\; 1\; t}} = {- \frac{a\; \Delta \; {Vn}}{m\left( {m + {2n}} \right)}}}} & {{Equation}\mspace{14mu} 28} \\{{I\; 0L_{{pref}\; 1\; t}} = {{{- I}\; 0R_{{pref}\; 1\; t}} = {- \frac{a^{2}\; \Delta \; V}{m + {2n}}}}} & {{Equation}\mspace{14mu} 29}\end{matrix}$

The phase A prefault current from the left side can be expressed as thesum of all three sequence currents:

$\begin{matrix}{{IAL}_{{pref}\; 1t} = \frac{{{m\; \Delta \; {V\left( {1 - a^{2}} \right)}} + {n\; \Delta \; {V\left( {1 - a} \right)}}}\;}{m\left( {m + {2n}} \right)}} & {{Equation}\mspace{14mu} 30}\end{matrix}$

FIG. 3 illustrates a simplified diagram of a sequence network 300 for aphase B pole-open condition showing a phase-A-to-ground fault network,according to one embodiment. FIG. 3 shows a positive-sequence loop 310with a transformer 312, a negative-sequence loop 320 with a transformer322, and a zero-sequence loop 330 with a transformer 332. The sequencecurrents for FIG. 3 can be expressed as:

$\begin{matrix}{{\begin{pmatrix}{{{ZL}\; 1} + {{ZS}\; 1} + {{ZR}\; 1}} & 0 & 0 & {{{- \left( {1 - d} \right)}{ZL}\; 1} - {{ZR}\; 1}} & a \\0 & {{{ZL}\; 2} + {{ZS}\; 2} + {{ZR}\; 2}} & 0 & {{{- \left( {1 - d} \right)}{ZL}\; 2} - {{ZR}\; 2}} & a^{2} \\0 & 0 & {{{ZL}\; 0} + {{ZS}\; 0} + {{ZR}\; 0}} & {{{- \left( {1 - d} \right)}{ZL}\; 0} - {{ZR}\; 0}} & 1 \\{{{ZS}\; 1} + {{dZL}\; 1}} & {{{ZS}\; 2} + {{dZL}\; 2}} & {{{ZS}\; 0} + {{dZL}\; 0}} & {3\; R} & 0 \\a^{2} & a & 1 & 0 & 0\end{pmatrix}\begin{pmatrix}{I\; 1\; L} \\{I\; 2\; L} \\{I\; 0\; L} \\{IF} \\{\left( \frac{1}{3} \right){VB}_{xy}}\end{pmatrix}} = \begin{pmatrix}{\Delta \; V} \\0 \\0 \\{VA} \\0\end{pmatrix}} & {{Equation}\mspace{14mu} 31} \\{\mspace{79mu} {{IF} = \frac{{{- 2}\; a^{2}m\; \Delta \; {V\left( {{{- 0.5}\; p} - q} \right)}} - {\Delta \; {{Vp}\left( {1 - a} \right)}\left( {m + {2\; n}} \right)} + {2\; {{mVA}\left( {m + {2\; n}} \right)}}}{{2\; {m\left( {{{- 0.5}\; p} - q} \right)}\left( {m_{1} + {2\; n_{1}}} \right)} - {3\; {{pm}_{1}\left( {m + {2\; n}} \right)}} + {6\; {{Rfm}\left( {m + {2\; n}} \right)}}}}} & {{Equation}\mspace{14mu} 32}\end{matrix}$

The Phase A fault current on the left side is equal to the sum of thethree sequence currents:

$\begin{matrix}{{IAL} = {\frac{\Delta \; {V\left( {m + n} \right)}}{m\left( {m + {2\; n}} \right)} + \frac{{- a}\; \Delta \; {Vn}}{m\left( {m + {2\; n}} \right)} - \frac{a^{2}\Delta \; V}{m + {2\; n}} + {\left( {{\left( {\frac{a^{2}}{2} + \frac{a}{2} - 1} \right)\left( \frac{m_{1} + {2\; n_{1}}}{m + {2\; n}} \right)} + \frac{m_{1}\left( {a^{2} - 2 + a} \right)}{2\; m}} \right){IF}}}} & {{Equation}\mspace{14mu} 33}\end{matrix}$

The sequence currents at both extremities of the lines are composed of aprefault term and a second term that is proportional to IF, giving theequations:

$\begin{matrix}{{I\; 0\; L} = {{I\; 0\; L_{preflt}} - {\frac{m_{1} + {2\; n_{1}}}{m + {2\; n}}{IF}}}} & {{Equation}\mspace{14mu} 34} \\{{I\; 2\; L} = {{I\; 2\; L_{preflt}} + {\left( {{\frac{a^{2}}{2}\frac{m_{1} + {2\; n_{1}}}{m + {2\; n}}} - \frac{m_{1}\left( {1 - a} \right)}{2\; m}} \right){IF}}}} & {{Equation}\mspace{14mu} 35} \\{{I\; 1\; L} = {{I\; 1\; L_{preflt}} + {\left( {{\frac{a}{2}\frac{m_{1} + {2\; n_{1}}}{m + {2\; n}}} - \frac{m_{1}\left( {a^{2} - 1} \right)}{2\; m}} \right){IF}}}} & {{Equation}\mspace{14mu} 36}\end{matrix}$

The left-side Phase A current can be expressed as:

$\begin{matrix}{{IAL} = {{IAL}_{preflt} + {\left( {{\left( {\frac{a^{2}}{2} + \frac{a}{2} - 1} \right)\left( \frac{m_{1} + {2\; n_{1}}}{m + {2\; n}} \right)} + \frac{m_{1}\left( {a^{2} - 2 + a} \right)}{2\; m}} \right){IF}}}} & {{Equation}\mspace{14mu} 37}\end{matrix}$

Given the identity,

a ² +a=−1  Equation 38

Equation 37 can be rewritten as:

$\begin{matrix}{{IAL} = {{IAL}_{preflt} - {\frac{3}{2}\left( {\frac{m_{1} + {2\; n_{1}}}{m + {2\; n}} + \frac{m_{1}}{m}} \right){IF}}}} & {{Equation}\mspace{14mu} 39}\end{matrix}$

Given that:

V1_(xy) +V2_(xy) +V0_(xy)=0  Equation 40

The voltages around the loop shown as the dashed line in FIG. 3 to drivethe equation:

V1L+V2L+V0L−d·ZL1·(I1L+I2L)

−d·ZL0·I0L·3Rf·IF=0  Equation 41

Adding and subtracting d·ZL1·I0L, results in the following:

V1L+V2L+V0L−d·ZL1·(I1L+I2L+I0L)

−d·I0L(ZL0−ZL1)I0L−3Rf·IF=0  Equation 42

Thus, if a phase-A-to-ground fault occurs during a phase B pole-opencondition, the relation between the voltage and current for the faultedloop, distance to the fault, fault resistance, and total sequencecurrent at the fault point is provided by:

VAL=d·ZL1·(IAL+K ₀ ·I0L)+3Rf·IF  Equation 43

In Equation 43, IF represents the total sequence current at the faultpoint in FIG. 3.

Equation 43 for the pole-open condition is similar to Equation 16 for anormal-mode (i.e., the three-pole closed) condition. As previouslystated, the sequence currents cannot be accurately calculated usingEquation 18. Instead, the relationship between the zero-sequence currentat the relay location and the total zero-sequence current at the faultpoint is given by:

$\begin{matrix}{{I\; 0\; L} = {\frac{{- a^{2}}\Delta \; V}{m + {2\; n}} - {\frac{m_{1} + {2\; n_{1}}}{m + {2\; n}}{IF}}}} & {{Equation}\mspace{14mu} 44}\end{matrix}$

The relationship between the negative-sequence current at the relay andthe total sequence current is expressed as:

$\begin{matrix}{{I\; 2\; L} = {\frac{{- a}\; \Delta \; {Vn}}{m\left( {m + {2\; n}} \right)} + {\left( {{\frac{a^{2}}{2}\frac{m_{1} + {2\; m_{1}}}{m + {2\; n}}} - \frac{m_{1}\left( {1 - a} \right)}{2\; m}} \right){IF}}}} & {{Equation}\mspace{14mu} 45}\end{matrix}$

Similarly, the relationship for the positive-sequence current is:

$\begin{matrix}{{I\; 1\; L} = {\frac{\Delta \; {V\left( {m + n} \right)}}{m\left( {m + {2\; n}} \right)} + {\left( {{\frac{a}{2}\frac{m_{1} + {2\; n_{1}}}{m + {2\; n}}} + \frac{m_{1}\left( {a^{2} - 1} \right)}{2\; m}} \right){IF}}}} & {{Equation}\mspace{14mu} 46}\end{matrix}$

The current distribution factors in a phase B pole-open condition may beexpressed as:

$\begin{matrix}{{C\; 0_{pob}} = {{{{C\; 0_{pob}}} \cdot e^{j\; \Psi_{0}}} = {- \frac{m_{1} + {2\; n_{1}}}{m + {2\; n}}}}} & {{Equation}\mspace{14mu} 47} \\{{C\; 2_{pob}} = {{{{C\; 2_{pob}}} \cdot e^{j\; \Psi_{2}}} = \left( {{\frac{a^{2}}{2}\frac{m_{1} + {2\; n_{1}}}{m + {2\; n}}} - \frac{m_{1}\left( {1 - a} \right)}{2\; m}} \right)}} & {{Equation}\mspace{14mu} 48} \\{{C\; 1_{pob}} = {{{{C\; 1_{pob}}} \cdot e^{j\; \Psi_{1}}} = \left( {{\frac{a}{2}\frac{m_{1} + {2\; n_{1}}}{m + {2\; n}}} + \frac{m_{1}\left( {a^{2} - 1} \right)}{2\; m}} \right)}} & {{Equation}\mspace{14mu} 49}\end{matrix}$

The total sequence current at the fault, as a function of thezero-sequence current change at the relay, is expressed as:

$\begin{matrix}{{IF} = {\frac{{I\; 0\; L} - {I\; 0\; L_{pref}}}{C\; 0_{pob}} = \frac{\Delta \; I\; 0\; L}{C\; 0_{pob}}}} & {{Equation}\mspace{14mu} 50}\end{matrix}$

The same can be done with respect to the positive-sequence andnegative-sequence current changes at the relay:

$\begin{matrix}{{IF} = {\frac{\Delta \; I\; 2\; L}{C\; 2_{pob}} = \frac{\Delta \; I\; 1\; L}{C\; 1_{pob}}}} & {{Equation}\mspace{14mu} 51}\end{matrix}$

By replacing IF in Equation 43 by its function of the change of thezero-sequence current at the relay, the following expression is derived:

$\begin{matrix}{{VAL} = {{{d \cdot {ZL}}\; {1 \cdot \left( {{IAL} + {{K_{0} \cdot I}\; 0\; L}} \right)}} + {3\; {{Rf} \cdot \frac{\Delta \; I\; 0\; L}{{{C\; 0_{pob}}} \cdot e^{j\; \Psi_{0}}}}}}} & {{Equation}\mspace{14mu} 52}\end{matrix}$

By multiplying both sides of Equation 52 by the conjugate of(ΔI0L·e^(−jψ) ⁰ ) and given that:

$\begin{matrix}{{{Im}\left( \frac{\Delta \; I\; 0\; {L \cdot {{conj}\left( {\Delta \; I\; 0\; {L \cdot e^{{- j}\; \Psi_{0}}}} \right)}}}{{{C\; 0_{pob}}} \cdot e^{j\; \Psi_{0}}} \right)} = {{{Im}\left( \frac{\Delta \; I\; 0\; L^{2}}{{C\; 0_{pob}}} \right)} = 0}} & {{Equation}\mspace{14mu} 53}\end{matrix}$

The distance to the fault can be expressed as:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {{VAL} \cdot {{conj}\left( {\Delta \; I\; 0\; {L \cdot e^{{- j}\; \Psi_{0}}}} \right)}} \right\rbrack}{{Im}\left\lbrack {{ZL}\; {1 \cdot I_{AG} \cdot {{conj}\left( {\Delta \; I\; 0\; {L \cdot e^{{- j}\; \Psi_{0}}}} \right)}}} \right\rbrack}} & {{Equation}\mspace{14mu} 54}\end{matrix}$

Alternatively, the distance to the fault can be expressed using thenegative-sequence current as the polarizing quantity as follows:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {{VAL} \cdot {{conj}\left( {\Delta \; I\; 2\; {L \cdot e^{{- j}\; \Psi_{2}}}} \right)}} \right\rbrack}{{Im}\left\lbrack {{ZL}\; {1 \cdot I_{AG} \cdot {{conj}\left( {\Delta \; I\; 2\; {L \cdot e^{{- j}\; \Psi_{2}}}} \right)}}} \right\rbrack}} & {{Equation}\mspace{14mu} 55}\end{matrix}$

Similarly, the distance to the fault can be expressed using thepositive-sequence current as the polarizing quantity as follows:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {{VAL} \cdot {{conj}\left( {\Delta \; I\; 1{L \cdot e^{{- j}\; \psi_{1}}}} \right)}} \right\rbrack}{{Im}\left\lbrack {{{ZL}\; {1 \cdot I_{AG}}}{\cdot {{conj}\left( {\Delta \; I\; 1{L \cdot e^{{- j}\; \psi_{1}}}} \right)}}} \right\rbrack}} & {{Equation}\mspace{14mu} 56}\end{matrix}$

Any of Equations 54, 55, and 56 can be used to calculate the distance tothe fault using data from one terminal (i.e., single-end faultlocation). The prefault sequence current can be used along with acompensating tilt angle. The current distribution factor relative toeach sequence current may be investigated to select the best choice toprovide the most accurate distance calculation.

FIG. 4 illustrates a diagram of a simplified single-line power system400 with a fault at a location d. The single-line power system ismodeled as a voltage VL 410 connected to a transmission line with animpedance ZS1 of 12.25∠84° and ZS0 of 38.5∠72° 411 between the powersource VL 410 and a left bus L 412. The transmission line may have animpedance ZL1 of 20.95∠86° and ZL0 of 75.32∠75.39° 413 between the leftbus L 412 and a right bus R 414. A fault location 413 a distance d fromthe left bus L 412 is shown as well. The transmission line may have animpedance ZR1 of 43.6∠80° and ZR0 of 128.62∠78° between the right bus R414 and a voltage VT 420.

The example power system 400 is a 60 km, 120 kV line with aphase-A-to-ground fault applied at a time t=100 ms at 66.66 percent ofthe line length with phase B open and a primary fault resistance of 50Ohms. FIGS. 5-11 relate to a comparison of single-ended fault locationcalculation using normal-mode approaches and the pole-open approachdescribed above. The graphs shown in FIGS. 5-10 are from a simulation ofvoltage and current waveforms acquired at a rate of 16 samples per cycle(960 Hz) and processed through a full-cycle cosine filter to calculatethe corresponding phasors.

FIG. 5 illustrates a graph 500 of a plurality of normal-mode currentdistribution factors I0 510 and I2 520 for the single-line power systemof FIG. 4. For the normal-mode calculations, a single tilt anglecorresponding to phase angles of C1 and C0 at the mid-distance, or d=0.5pu. At d=0.5 pu, the angles chosen may be:

φ₀=1.1721°

φ₂=−1.1132°

Using these tilt angles and introducing these values for the normal-modecalculations, the time loci of the distance to the fault can becalculated using either the negative-sequence or zero-sequence currentsas the polarizing quantities.

FIG. 6 illustrates distance calculation errors when using normal-modedistance calculations with a zero-sequence I0 (solid line) and anegative-sequence I2 (dashed line) polarization during the open-polefault condition in the single-line power system of FIG. 4.

FIG. 6 shows that the zero-sequence polarization provides an error ofabout 46 percent (0.36 pu instead of 0.666 pu for the distance [d]) andthe negative-sequence polarization ends up with an error of 41 percent(fault location at 0.94 pu instead of 0.666 pu).

FIG. 7 illustrates that the reason that the normal-mode distancecalculation approaches in FIG. 6 for the single-line power system ofFIG. 4 are inaccurate is because the pre-fault zero-sequence andnegative-sequence currents are non-negligible. Specifically, FIG. 7shows that both the zero-sequence and negative-sequence currents aresignificant before the fault at about 100 ms.

Thus, the normal-mode approach to calculating a distance to a fault mayprovide an erroneous distance calculation for a pole-open transmissionline system. Accordingly, in various embodiments, a system fordetermining a fault location may calculate a distance using thedual-sequence, pole-open approaches described herein.

FIG. 8 illustrates the variation of a plurality of current distributionfactor phase angles 800 for each of a zero-sequence I0, apositive-sequence I1, and a negative-sequence I2 current for theopen-pole fault condition in FIG. 4. In the illustrated embodiment, itis readily apparent that the positive-sequence current I1 provides thebest result because it has the least variation of the phase angle overthe distance range of 0 to 1 pu, followed by the zero-sequence currentI0. The last choice would be the negative-sequence current I2 because ithas the largest variation of the phase angles. In each case, a tiltangle can be selected that corresponds to the mid-range distance valueof 0.5 pu, or:

ψ₁=−0.0836°

ψ₂=−1.1826°

ψ₃=0.8721°

FIG. 9 illustrates a single-ended fault location calculation 900following a phase B pole-open condition using the pole-open,two-sequence calculation approach described herein. The zero-sequencecurrent polarization provides the best result, although thepositive-sequence current polarization fault location might also proveto be an adequate selection. The polarization using thenegative-sequence current provides the worst fault location and relativeaverage error, although it is still below 2.3 percent (0.6807 pumeasured against the true value of 0.666 pu). The zero-sequence currentpolarization has an average relative error of 0.3 percent (0.668 pumeasured). The positive-sequence current polarization has an averageerror of 0.9 percent (0.672 pu measured). All three results are muchbetter than the normal-mode calculations.

The phase angle of the negative-sequence current distribution factor asa function of the distance, d, to the fault can be curve-fitted withhigh accuracy using a third order polynomial:

ψ₂(d)=−0.5015·d ³−0.4387·d ²−1.9232·d−0.0475  Equation 57

The distance to the fault can also be computed by representing the tiltangle as a function of the distance to the fault:

$\begin{matrix}{d = \frac{{Im}\left\lbrack {{VAL} \cdot {{conj}\left( {\Delta \; I\; 2L} \right)} \cdot e^{{- j}\; {\psi_{2}{(d)}}}} \right\rbrack}{{Im}\left\lbrack {{{ZL}\; {1 \cdot I_{AG}}}{\cdot {{conj}\left( {\Delta \; I\; 2L} \right)} \cdot e^{{- j}\; {\psi_{2}{(d)}}}}} \right\rbrack}} & {{Equation}\mspace{14mu} 58}\end{matrix}$

In some embodiments, an iterative process is utilized that starts with atilt angle of zero to iteratively calculate the distance, d. Thedistance, d, can then be used to compute a new tilt angle using thepolynomial curve-fit equation above. The process is iteratively repeateduntil the error is within a predefined tolerance.

FIG. 10 illustrates a graphical representation 1000 of a fault locationcalculation following a phase B pole-open condition using an iterativedual-sequence network calculation approach with a polynomialapproximation of the tilt angle. As illustrated, the iterative approachallows a distance to be calculated with practically no error.

It is appreciated that in some instances the iterative fault locationmethod may not be practical as it utilizes a precise knowledge of thenetwork parameters and particularly the source impedances. While thismay not always be possible, the example shows the factors impacting thesingle-ended fault location and shows what is possible with an idealsingle-ended fault location method.

FIG. 11 illustrates a simplified diagram of a negative-sequence network1100, according to one embodiment. The network includes a line 1112 witha left side impedance ZS2 1120, a left side fault impedance 1130, afault resistance 1135, a right side fault impedance 1140, and a rightside impedance 1160. Using data from both the right and left side, adoubled-ended fault location approach may be utilized.

Double-ended fault location may be applied by using thenegative-sequence network in normal mode. Assuming synchronous samplingat both line terminals and given that the voltage drop between the leftand right bus and VF is the same, the following expression can be used:

V2L−I2L·d·ZL1=V2R−(1−d)·ZL1·I2R  Equation 59

An equation for the fault location, d, assuming that all poles areclosed can be expressed as:

$\begin{matrix}{d = \frac{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; {1 \cdot I}\; 2R}}{{ZL}\; {1 \cdot \left( {{I\; 2L} + {I\; 2R}} \right)}}} & {{Equation}\mspace{14mu} 60}\end{matrix}$

A similar equation for the pole-open configuration with phase Bpole-open, as illustrated in the negative-sequence network shown in FIG.3, can be expressed as:

$\begin{matrix}{d = \frac{\begin{matrix}{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {{a \cdot I}\; 1R}} \right)} -} \\{a\left( {{V\; 1L} - {V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{I\; 2L} - {I\; 2R}} \right)} - {{a \cdot {ZL}}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}} & {{Equation}\mspace{14mu} 61}\end{matrix}$

As illustrated in Equation 62, the distance for the pole-open conditioninvolves voltages and currents from two sequence networks. Specifically,the distance for the pole-open condition includes both thepositive-sequence and negative-sequence networks. The double-ended faultlocation during a phase A pole-open condition can be expressed as:

$\begin{matrix}{d = \frac{\begin{matrix}{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {I\; 1R}} \right)} -} \\\left( {{V\; 1L} - {V\; 1R}} \right)\end{matrix}}{{{ZL}\; 1\left( {{I\; 2L} - {I\; 2R}} \right)} - {{ZL}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}} & {{Equation}\mspace{14mu} 62}\end{matrix}$

Similarly, the double-ended fault location method for a phase Cpole-open condition can be expressed as:

$\begin{matrix}{d = \frac{\begin{matrix}{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {{a^{2} \cdot I}\; 1R}} \right)} -} \\{a^{2}\left( {{V\; 1L} - {V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{I\; 2L} - {I\; 2R}} \right)} - {{a^{2} \cdot {ZL}}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}} & {{Equation}\mspace{14mu} 63}\end{matrix}$

In Equations 61, 62, and 63, a represents the conventional complexoperator 1∠120°.

By using a generic complex operator, α, that represents a 1 for a phaseA open-pole, 1∠120° for a phase B open-pole, and (1∠120°)² for a phase Copen-pole, the dual sequence, double-ended fault location algorithm canbe expressed in two ways. A first way is represented below by Equation64:

$\begin{matrix}{d = \frac{\begin{matrix}{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {{\alpha I}\; 1R}} \right)} -} \\{\alpha \left( {{V\; 1L} - {V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{I\; 2L} - {I\; 2R}} \right)} - {{\alpha ZL}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}} & {{Equation}\mspace{14mu} 64}\end{matrix}$

Alternatively, the distance calculation may be represented by theincremental equation below:

$\begin{matrix}{d = \frac{\begin{matrix}{\left( {{\Delta \; V\; 2L} - {\Delta \; V\; 2R}} \right) + {{ZL}\; 1\left( {{\Delta \; I\; 2R} - {{\alpha\Delta}\; I\; 1R}} \right)} -} \\{\alpha \left( {{\Delta \; V\; 1L} - {\Delta \; V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{\Delta \; I\; 2L} - {\Delta \; I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{\Delta \; I\; 1L} + {\Delta \; I\; 1R}} \right)}}} & {{Equation}\mspace{14mu} 65}\end{matrix}$

In Equations 65, the distance can be found using incremental positive(or negative) sequence voltage and/or current phasors. The incrementalvoltage or current phasor is the voltage or current phasor during thefault after subtracting the same voltage or current phasor before thefault. In each of these situations, a pole is open before the faultoccurs. Accordingly, the negative sequence voltage or current phasorwill not be a zero phasor. Therefore, using “preflt” to signify apre-fault state, the following are used in Equations 65:

ΔV2L=V2L−V2L _(preflt)

ΔI2L=I2L−I2L _(preflt)

ΔV1L=V1L−V1L _(preflt)

ΔI1L=I1L−I1L _(preflt)

ΔV2R=V2R−V2R _(preflt)

ΔI2R=I2R−I2R _(preflt)

ΔV1R=V1R−V1R _(preflt)

ΔI1R=I1R−I1R _(preflt)

FIG. 12 illustrates a graphical representation 1200 of a fault locationcalculation of the single-line power system in FIG. 4 using adouble-ended normal-mode approach and the double-ended pole-openapproach (two-sequence approach). The illustration assumes a faultresistance of 20 Ohms on a 120 kV system with a phase-C-to-ground faultwith phase B open at 66 percent of the line length. Given that theactual fault distance is 0.666 pu, it is readily apparent from FIG. 12that the pole-open, dual-sequence approach is far superior to thenormal-mode approach for pole-open fault conditions.

FIG. 13 illustrates a close-up view 1300 of FIG. 12 to better illustratethe accuracy of the pole-open, dual-sequence approach. The relativeerror is approximately 0.5 percent for the illustrated example.

FIG. 14 illustrates another example diagram of a single-line powersystem 1400 with a fault at a location d, according to one embodiment.The single-line power system is modeled as a voltage VL 1410 connectedto a transmission line with an impedance ZS1 of 15∠88° and ZS0 of52.2∠73.3° 1411 between the power source VL 1410 and a left bus L 1412.The transmission line may have an impedance ZL1 of 73∠87.27° and ZL0 of274.1∠82.14° 1414 between the left bus L 1412 and a right bus R 1415. Afault location 1413 a distance d from the left bus L 1412 is shown aswell. The transmission line may have an impedance ZR1 of 15∠88° and ZR0of 52.2∠73.3° between the right bus R 1415 and a voltage VT 1417.

The example power system 1400 is a 200 km, 500 kV line with aphase-C-to-ground fault applied at a time t=100 ms at 66.66 percent ofthe line length with phase B open and a primary fault resistance of 20Ohms. FIGS. 5-11 relate to a comparison of single-ended fault locationcalculation using normal-mode approaches and the pole-open approachdescribed above. The graphs shown in FIGS. 5-10 are from a simulation ofvoltage and current waveforms acquired at a rate of 16 samples per cycle(960 Hz) and processed through a full-cycle cosine filter to calculatethe corresponding phasors.

FIG. 15 illustrates a graphical representation 1500 showing the improvedaccuracy of the double-ended pole-open, dual-sequence approach ascompared to the double-ended normal-mode approach for the system inwhich a fault occurs with phase C open.

FIG. 16 illustrates a close-up view 1600 of FIG. 15 to better illustratethe absolute accuracy of the pole-open, dual-sequence approach. Again,the relative error is less than 0.5 percent for the illustrated example.

FIG. 17 illustrates a flow chart of a method 1700 for dual-sequence,single-ended fault location calculation, according to one embodiment.Many of the steps may be implemented in any order. However, asillustrated, a first step may include determining, at 1702, faultcurrent value, I_(AG), of a faulted loop during a fault. A prefaultsequence current value, IXL_(pref), may be calculated 1704, for thethree-phase transmission system during the pole-open condition.Subsequently, a system may calculate 1706 a sequence fault currentvalue, IXL, during a fault. An incremental value of the sequencecurrent, ΔIXL, may be calculated 1708 that corresponds to the differencebetween the prefault sequence current value, IXL_(pref), and thesequence fault current value, IXL.

The system may then identify 1710 a current distribution factor phaseangle, e^(−jψx), based on impedance values of at least two sequencenetworks (e.g., a positive-sequence network, a negative-sequencenetwork, and/or a zero-sequence network).

The system may then calculate 1712 a tilt value corresponding to aconjugate of a product of the incremental value of the sequence current,ΔIXL, and the current distribution factor phase angle, e^(−jψx). Adistance to the fault may then be determined 1714 based on the quotientof (a) an imaginary portion of a product of the voltage of the faultedloop, VAL, and the tilt value divided by (b) an imaginary portion of aproduct of a positive-sequence impedance, ZL1, the fault current value,I_(AG), and the tilt value.

FIG. 18 illustrates a flow chart of a method 1800 for dual-sequence,double-ended fault location calculation, according to one embodiment. Asystem may be configured to identify 1802 which one of three phases wasin a pole-open condition during a fault between a first relay and asecond relay. A complex operator, α, may be selected 1804, where thecomplex operator, α, is equal to 1 for a phase A open-pole, 1∠120° for aphase B open-pole, and (1∠120°)² for a phase C open-pole. An impedancevalue ZL1 may be calculated or measured, at 1806.

While variations of the exact order and variables may be possible, theillustrated method shows a plurality of differences (and sums) betweennegative- and positive-sequence voltages and currents 1808, including: adifference between a negative-sequence voltage, V2L, at the first relayand a negative-sequence voltage, V2R, at the second relay; a differencebetween a negative-sequence current, I2R, at the second relay and apositive-sequence current, I1 R, at the second relay; a differencebetween a positive-sequence voltage, V1L, at the first relay and apositive-sequence voltage, V1R at the second relay; a sum of anegative-sequence current, I2L, at the first relay and anegative-sequence current, I2R, at the second relay; and a sum of thepositive-sequence current, I1L, at the first relay and apositive-sequence current, I1R, at the second relay.

Finally, a distance to the fault may be determined 1810 by implementingan algorithm that satisfies an equation:

$d = {\frac{\begin{matrix}{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( \; {{I\; 2R} - {\alpha \; I\; 1R}} \right)} -} \\{\alpha\left( \; {{V\; 1L} - {V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{I\; 2L} - {I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{I\; 1L} + \; {I\; 1R}} \right)}}.}$

Alternatively, the distance to the fault may be determined byimplementing an algorithm that satisfies an equation:

$d = {\frac{\begin{matrix}{\left( {{\Delta \; V\; 2L} - {\Delta \; V\; 2R}} \right) + {{ZL}\; 1\left( {{\Delta \; I\; 2R} - {{\alpha\Delta}\; I\; 1R}} \right)} -} \\{\alpha \left( {{\Delta \; V\; 1L} - {\Delta \; V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{\Delta \; I\; 2L} + {\Delta \; I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{\Delta \; I\; 1L} + {\Delta \; I\; 1R}} \right)}}.}$

Thus, systems and/or methods may be constructed that calculate adistance to a fault based on double-ended data that implements adistance calculation algorithm that satisfies at least one of (1)

$d = {\frac{\left( \; {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {\alpha \; I\; 1R}} \right)} - {\alpha\left( \; {{V\; 1L} - {V\; 1R}} \right)}}{{{ZL}\; 1\left( \; {{I\; 2L} + \; {I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}\mspace{14mu} {and}\; {\quad{{d = \frac{\left( {{\Delta \; V\; 2L} - {\Delta \; V\; 2R}} \right) + {{ZL}\; 1\left( {{\Delta \; I\; 2R} - {{\alpha\Delta}\; I\; 1R}} \right)} - {\alpha \left( {{\Delta \; V\; 1L} - {\Delta \; V\; 1R}} \right)}}{{{ZL}\; 1\left( {{\Delta \; I\; 2L} + {\Delta \; I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{\Delta \; I\; 1L} + {\Delta \; I\; 1R}} \right)}}},}}}$

as described above.

FIG. 19 illustrates an IED 1900 for performing single-ended ordouble-ended fault location calculations, according to variousembodiments. The IED includes a bus 1920 connecting a processor 1930 orprocessing unit(s) to a memory 1940, a network interface 1950, and acomputer-readable storage medium 1970. The computer-readable storagemedium 1970 may include or interface with software, hardware, orfirmware modules for implementing various portions of the systems andmethods described herein. The separation of the modules is merely anexample and any combination of the modules or further division may bepossible.

A current value module 1980 may include or interface with one or morecurrent sensors and be configured to determine a current value before,after, or during a fault event. A sequence module 1982 may be configuredto calculate negative-, positive-, and/or zero-sequence currents. Adistribution factor module 1984 may be configured to determine a currentdistribution factor phase angle, e^(−jψx), based on impedance values ofat least two sequence networks. A tilt module 1986 may be configured tocalculate a tilt value corresponding to a conjugate of a product of theincremental value of the sequence current, ΔIXL, and the currentdistribution factor phase angle, e^(−jψx).

A dividend module 1988 may be configured to calculate a distancedividend corresponding to an imaginary portion of a product of thevoltage of the faulted loop, VAL, and the tilt value. A divisor module1990 may be configured to calculate a distance divisor corresponding toan imaginary portion of a product of a positive-sequence impedance, ZL1,the fault current value, I_(AG), and the tilt value. A distancedetermination or calculation module 1992 may be configured to determinea distance to the fault corresponding to a quotient of the distancedividend and the distance divisor.

A pole-open identification module 1994 may be configured to identifywhich one of three phases was in a pole-open condition during a faultbetween the first relay and the second relay. A complex operator module1996 may be configured to identify a complex operator, α, correspondingto the pole-open phase, where the complex operator, α, is equal to 1 fora phase A open-pole, 1∠120° for a phase B open-pole, and (1∠120°)² for aphase C open-pole. Finally, an impedance module 1998 may be configuredto determine an impedance, ZL1, between the first relay and the secondrelay.

This disclosure has been made with reference to various embodiments,including the best mode. However, those skilled in the art willrecognize that changes and modifications may be made to the embodimentswithout departing from the scope of the present disclosure. While theprinciples of this disclosure have been shown in various embodiments,many modifications of structure, arrangements, proportions, elements,materials, and components may be adapted for a specific environmentand/or operating requirements without departing from the principles andscope of this disclosure. These and other changes or modifications areintended to be included within the scope of the present disclosure.

This disclosure is to be regarded in an illustrative rather than arestrictive sense, and all such modifications are intended to beincluded within the scope thereof. Likewise, benefits, other advantages,and solutions to problems have been described above with regard tovarious embodiments. However, benefits, advantages, solutions toproblems, and any element(s) that may cause any benefit, advantage, orsolution to occur or become more pronounced are not to be construed as acritical, required, or essential feature or element. The scope of thepresent invention should, therefore, be determined by the followingclaims:

What is claimed:
 1. A single-ended fault location determination systemfor calculating a distance to a fault during a pole-open condition of athree-phase transmission system, comprising: a current value modulecomprising at least one current sensor to determine: current values onat least two phase lines of a three-phase transmission system during apole-open condition, and a fault current value, I_(AG), of a faultedloop during a fault; a sequence module in communication with the currentvalue module to determine: a prefault sequence current value,IXL_(pref), of the three-phase transmission system during the pole-opencondition, and a sequence fault current value, IXL, during a fault,wherein an incremental value of the sequence current, ΔIXL, correspondsto the difference between the prefault sequence current value,IXL_(pref), and the sequence fault current value, IXL; a distributionfactor module to determine a current distribution factor phase angle,e^(−jψx), based on impedance values of at least two sequence networks; atilt module to calculate a tilt value corresponding to a conjugate of aproduct of the incremental value of the sequence current, ΔIXL, and thecurrent distribution factor phase angle, e^(−jψx); a dividend module tocalculate a distance dividend corresponding to an imaginary portion of aproduct of the voltage of the faulted loop, VAL, and the tilt value; adivisor module to calculate a distance divisor corresponding to animaginary portion of a product of a positive-sequence impedance, ZL1,the fault current value, I_(AG), and the tilt value; and a distancedetermination module to determine a distance to the fault correspondingto a quotient of the distance dividend and the distance divisor.
 2. Thesystem of claim 1, wherein the prefault sequence current value,IXL_(pref), comprises a prefault zero-sequence current value I0L_(pref),and wherein the sequence fault current value, IXL, comprises azero-sequence current value I0L, such that the incremental value of thesequence current, ΔIXL, comprises an incremental value of thezero-sequence current ΔI0L, and the current distribution factor phaseangle, e^(−jψx), is a zero-sequence current distribution factor phaseangle, e^(−jψ0).
 3. The system of claim 1, wherein the prefault sequencecurrent value, IXL_(pref), comprises a prefault positive-sequencecurrent value I1L_(pref), and wherein the sequence fault current value,IXL, comprises a positive-sequence current value I1L, such that theincremental value of the sequence current, ΔIXL, comprises anincremental value of the positive-sequence current ΔI1L, and the currentdistribution factor phase angle, e^(−jψx), is the positive-sequencecurrent distribution factor phase angle, e^(−jψ1).
 4. The system ofclaim 1, wherein the prefault sequence current value, IXL_(pref),comprises a prefault negative-sequence current value I2L_(pref), andwherein the sequence fault current value, IXL, comprises anegative-sequence current value I2L, such that the incremental value ofthe sequence current, ΔIXL, comprises an incremental value of thenegative-sequence current ΔI2L, and the current distribution factorphase angle, e^(−jψx), is the negative-sequence current distributionfactor phase angle, e^(−jψ2).
 5. The system of claim 1, wherein thecurrent distribution factor phase angle, e^(−jψx), is based on impedancevalues of a positive-sequence network of the three-phase transmissionsystem and a zero-sequence network of the three-phase transmissionsystem.
 6. The system of claim 1, wherein the current distributionfactor phase angle, e^(−jψx), is based on impedance values of anegative-sequence network of the three-phase transmission system and azero-sequence network of the three-phase transmission system.
 7. Thesystem of claim 1, wherein the current distribution factor phase angle,e^(−jψx), is based on impedance values of a negative-sequence network ofthe three-phase transmission system and a positive-sequence network ofthe three-phase transmission system.
 8. The system of claim 1, whereinthe current distribution factor phase angle, e^(−jψx), is further made afunction of the distance, d, to the fault and wherein the distance, d,is solved through an iterative polynomial curve-fitting algorithm. 9.The system of claim 1, wherein the voltage of the faulted loop, VAL, ismeasured using a voltage meter.
 10. A method for calculating a distanceto a fault during a pole-open condition of a three-phase transmissionsystem from a single end, comprising: determining, via at least onecurrent sensor, a fault current value, I_(AG), of a faulted loop duringa fault; calculating a prefault sequence current value, IXL_(pref), ofthe three-phase transmission system during the pole-open condition, andcalculating a sequence fault current value, IXL, during a fault,calculating an incremental value of the sequence current, ΔIXL,corresponding to the difference between the prefault sequence currentvalue, IXL_(pref), and the sequence fault current value, IXL;identifying a current distribution factor phase angle, e^(−jψx), basedon impedance values of at least two sequence networks selected from thegroup of sequence networks consisting of a positive-sequence network, anegative-sequence network, and a zero-sequence network; calculating atilt value corresponding to a conjugate of a product of the incrementalvalue of the sequence current, ΔIXL, and the current distribution factorphase angle, e^(−jψx); determining a distance to the fault based on thequotient of (a) an imaginary portion of a product of the voltage of thefaulted loop, VAL, and the tilt value divided by (b) an imaginaryportion of a product of a positive-sequence impedance, ZL1, the faultcurrent value, I_(AG), and the tilt value.
 11. A method for calculatinga distance to a fault during a pole-open condition of a three-phasetransmission system between two relays, comprising: identifying whichone of three phases was in a pole-open condition during a fault betweena first relay and a second relay; selecting a corresponding complexoperator, α, where the complex operator, α, is equal to 1 for a phase Aopen-pole, 1∠120° for a phase B open-pole, and (1∠120°)² for a phase Copen-pole; determining an impedance value ZL1 between the first relayand the second relay; calculating a difference between anegative-sequence voltage, V2L, at the first relay and anegative-sequence voltage, V2R, at the second relay; calculating adifference between a negative-sequence current, I2R, at the second relayand a positive-sequence current, I1R, at the second relay; calculating adifference between a positive-sequence voltage, V1L, at the first relayand a positive-sequence voltage, V1R, at the second relay; calculating asum of a negative-sequence current, I2L, at the first relay and anegative-sequence current, I2R, at the second relay; calculating a sumof the positive-sequence current, I1L, at the first relay and apositive-sequence current, I1R, at the second relay; and determining adistance to the fault by implementing an algorithm that at leastapproximately satisfies at least one of two distance equations:$\begin{matrix}{{d = \frac{\left( \; {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {\alpha \; I\; 1R}} \right)} - {\alpha\left( \; {{V\; 1L} - {V\; 1R}} \right)}}{{{ZL}\; 1\left( \; {{I\; 2L} - \; {I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}}\;,{and}} & (1) \\{d = {\frac{\begin{matrix}{\left( {{\Delta \; V\; 2L} - {\Delta \; V\; 2R}} \right) + {{ZL}\; 1\left( {{\Delta \; I\; 2R} - {{\alpha\Delta}\; I\; 1R}} \right)} -} \\{\alpha \left( {{\Delta \; V\; 1L} - {\Delta \; V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{\Delta \; I\; 2L} + {\Delta \; I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{\Delta \; I\; 1L} + {\Delta \; I\; 1R}} \right)}}.}} & (2)\end{matrix}$
 12. The method of claim 10, wherein phase B is in apole-open condition during the fault such that α is equal to 1∠120°. 13.The method of claim 10, further comprising: reporting the calculateddistance to the fault as a range based on an estimated error percentage.14. A non-transitory computer-readable medium with instructions storedthereon that, when executed by a processor, cause an intelligentelectronic device (IED) to perform operations for calculating a distanceto a fault during a pole-open condition of a three-phase transmissionsystem between two relays, the operations comprising: receiving datafrom at least one of a first relay and a second indicating which one ofthree phases was in a pole-open condition during a fault between thefirst relay and the second relay; selecting a corresponding complexoperator, α, where the complex operator, α, is equal to 1 for a phase Aopen-pole, 1∠120° for a phase B open-pole, and (1∠120°)² for a phase Copen-pole; identifying an impedance value ZL1 between the first relayand the second relay; calculating a difference between anegative-sequence voltage, V2L, at the first relay and anegative-sequence voltage, V2R, at the second relay; calculating adifference between a negative-sequence current, I2R, at the second relayand a positive-sequence current, I1R, at the second relay; calculating adifference between a positive-sequence voltage, V1L, at the first relayand a positive-sequence voltage, V1R, at the second relay; calculating asum of a negative-sequence current, I2L, at the first relay and anegative-sequence current, I2R, at the second relay; calculating a sumof the positive-sequence current, I1L, at the first relay and apositive-sequence current, I1R, at the second relay; and determining adistance to the fault by implementing an algorithm that at leastapproximately satisfies at least one of two distance equations:$\begin{matrix}{{d = \frac{\left( \; {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {\alpha \; I\; 1R}} \right)} - {\alpha\left( \; {{V\; 1L} - {V\; 1R}} \right)}}{{{ZL}\; 1\left( {{I\; 2L} - \; {I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}},{and}} & (1) \\{d = {\frac{\begin{matrix}{\left( {{\Delta \; V\; 2L} - {\Delta \; V\; 2R}} \right) + {{ZL}\; 1\left( {{\Delta \; I\; 2R} - {{\alpha\Delta}\; I\; 1R}} \right)} -} \\{\alpha \left( {{\Delta \; V\; 1L} - {\Delta \; V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{\Delta \; I\; 2L} + {\Delta \; I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{\Delta \; I\; 1L} + {\Delta \; I\; 1R}} \right)}}.}} & (2)\end{matrix}$
 15. An intelligent electronic device for performing adouble-ended fault location determination during a pole-open conditionof a three-phase transmission system, comprising a pole-openidentification module to identify which one of three phases was in apole-open condition during a fault between the first relay and thesecond relay; a complex operator determination module to identify acomplex operator, α, corresponding to the pole-open phase, where thecomplex operator, α, is equal to 1 for a phase A open-pole, 1∠120° for aphase B open-pole, and (1∠120°)² for a phase C open-pole; an impedancemodule to determine an impedance, ZL1, between the first relay and thesecond relay; a processing unit to: calculate a difference between anegative-sequence voltage, V2L, at the first relay and anegative-sequence voltage, V2R, at the second relay; calculate adifference between a negative-sequence current, I2R, at the second relayand a positive-sequence current, I1R, at the second relay; calculate adifference between a positive-sequence voltage, V1L, at the first relayand a positive-sequence voltage, V1R, at the second relay; calculate asum of a negative-sequence current, I2L, at the first relay and anegative-sequence current, I2R, at the second relay; calculate a sum ofthe positive-sequence current, I1L, at the first relay and apositive-sequence current, I1R, at the second relay; and determine adistance to the fault by implementing an algorithm that at leastapproximately satisfies at least one of two distance equations:$\begin{matrix}{{d = \frac{\left( {{V\; 2L} - {V\; 2R}} \right) + {{ZL}\; 1\left( {{I\; 2R} - {\alpha \; I\; 1R}} \right)} - {\alpha\left( \; {{V\; 1L} - {V\; 1R}} \right)}}{{{ZL}\; 1\left( \; {{I\; 2L} - \; {I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{I\; 1L} + {I\; 1R}} \right)}}},{and}} & (1) \\{d = {\frac{\begin{matrix}{\left( {{\Delta \; V\; 2L} - {\Delta \; V\; 2R}} \right) + {{ZL}\; 1\left( {{\Delta \; I\; 2R} - {{\alpha\Delta}\; I\; 1R}} \right)} -} \\{\alpha \left( {{\Delta \; V\; 1L} - {\Delta \; V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{\Delta \; I\; 2L} + {\Delta \; I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{\Delta \; I\; 1L} + {\Delta \; I\; 1R}} \right)}}.}} & (2)\end{matrix}$
 16. The system of claim 15, wherein the poleidentification module comprises a sensor configured to measure one of avoltage, a resistance, and a current to measure which of the threephases is in a pole-open condition.
 17. The system of claim 15, whereinthe impedance module is configured to measure the impedance, ZL1,between the first relay and the second relay.
 18. The system of claim15, wherein the impedance module is configured to calculate theimpedance, ZL1, between the first relay and the second relay based onstored information regarding the three-phase transmission system. 19.The system of claim 15, wherein the processing unit is furtherconfigured to calculate the voltage across the open pole of the phase inthe open-pole condition.
 20. The system of claim 15, wherein phase C isin an open-pole condition such that the complex operator, α, is equal to(1∠120°)².
 21. The system of claim 15, wherein the implemented algorithmsatisfies the distance equation ${d = \frac{\begin{matrix}{\left( {{\Delta \; V\; 2L} - {\Delta \; V\; 2R}} \right) + {{ZL}\; 1\left( {{\Delta \; I\; 2R} - {{\alpha\Delta}\; I\; 1R}} \right)} -} \\{\alpha \left( {{\Delta \; V\; 1L} - {\Delta \; V\; 1R}} \right)}\end{matrix}}{{{ZL}\; 1\left( {{\Delta \; I\; 2L} + {\Delta \; I\; 2R}} \right)} - {\alpha \; {ZL}\; 1\left( {{\Delta \; I\; 1L} + {\Delta \; I\; 1R}} \right)}}},$and wherein each delta (Δ) value represents an incremental differencebetween a fault value and a pre-fault value.